The answer is 21 triangles.
The figure is a fan: 7 straight rays spreading from a single apex point at the bottom, all cut by one straight horizontal line near the top. Here’s the key insight that makes this count provable instead of guessy: every triangle in the fan is formed by picking any 2 of the 7 rays plus the horizontal line at the top. Two rays make the sides, the horizontal line closes the top — that’s a triangle, every single time. So the count is simply: how many pairs can you pick from 7 rays?
Count them width by width, starting from the leftmost ray and widening one step at a time — a triangle’s width is how many gaps sit between its two rays:
- Width 1 (neighboring rays): 6 triangles
- Width 2: 5 triangles
- Width 3: 4 triangles
- Width 4: 3 triangles
- Width 5: 2 triangles
- Width 6 (the two outermost rays — the whole fan): 1 triangle
Now add them up: 6 + 5 + 4 + 3 + 2 + 1 = 21. (Mathematicians write the same thing as C(7,2) = 21 — the number of ways to choose 2 rays from 7.)
See why the staircase works? Each wider width has exactly one fewer place to sit than the width before it, so the counts step down 6, 5, 4, 3, 2, 1 with no gaps and no double-counting. The two traps are stopping at the skinny triangles (most people find the 6 narrow ones and quit) and forgetting the biggest one of all: the entire fan, outermost ray to outermost ray, is a triangle too — and it’s worth exactly 1.
Did you land on 21, or did the wide triangles at the edges escape your count? Be honest in the comments — and watch our page for tomorrow’s counting game.
